THE STABLE SYMPLECTIC CATEGORY AND GEOMETRIC QUANTIZATION

NITU KITCHLOO

ABSTRACT. We study a stabilization of the symplectic category introduced by A. Weinsteinas a domain for the geometric quantization functor. The symplectic category is a topologi-cal category with objects given by symplectic manifolds, and morphisms being suitable la-grangian correspondences. The main drawback of Weinstein’s symplectic category is thatcomposition of morphisms cannot always be defined. Our stabilization procedure rectifiesthis problem while remaining faithful to the original notion of composition. The stablesymplectic category is enriched over the category of spectra (in particular, its morphismscan be described as infinite loop spaces of stable lagrangians), and it possesses several ap-pealing properties that are relevant in the context of geometric quantization.

CONTENTS

1. Introduction 1

2. Stable lagrangian immersions and their moduli space 3

3. The Stable Symplectic category 5

4. Internal structure of the stable symplectic category 8

5. The Metaplectic Category 9

6. Symbols and Thom classes 11

7. A second Stabilization 16

8. Appendix: Some computations and Questions 19

References 21

1. INTRODUCTION

Motivated by earlier work by Guillemin and Sternberg [8], A. Weinstein [10, 12] intro-duced the symplectic category as a domain category for constructing the (yet to be com-pletely defined) geometric quantization functor. The objects of this topological categoryare symplectic manifolds, and morphisms between two symplectic manifolds (M, ω) and(N, η) are defined as lagrangian correspondences inside M×N , where the conjugate sym-plectic manifold M is defined by the pair (M,−ω). Geometric quantization is an attemptat constructing a canonical representation of this category. There are many categoriesclosely related to the symplectic category (for example the Fukaya 2-category), that weshall not consider in this document.

Date: May 6, 2012.Nitu Kitchloo is supported in part by NSF through grant DMS 1005391.

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Composition of two lagrangians L1 ⊂ M ×N and L2 ⊂ N ×K in the symplectic categoryis defined as L1 ∗ L2 ⊂ M × K given by the image of the projection

π1,4 : M × N × N × K −→ M × K,

restricted to the intersection of the two subspaces: L1 × L2 and M × ∆(N) × K, where∆(N) ⊂ N × N is the diagonal submanifold. The first observation to make is that thesymplectic category is not strictly a category because composition as defined above doesnot always yield a lagrangian submanifold. For composition to yield a lagrangian sub-manifold, the intersection that is used to define it must be transverse. One way to fix thisproblem was introduced by Wehrheim and Woodward [13] where they consider the freecategory generated by all the correspondences modulo the obvious relation if the pair ofcomposable morphisms is transverse.

In this document, we introduce another method of extending the symplectic categoryinto an honest category. We introduce a moduli space of stable lagrangian immersionsin a symplectic manifold of the form M × N . This moduli space can be described as theinfinite loop space corresponding to a certain Thom spectrum. Taking this as the spaceof morphisms defines the stable symplectic category that is naturally enriched over themonoidal category of spectra (under smash product). Composition in this stable sym-plectic category remains faithful to the original definition introduced by Weinstein. Inaddition, it factors through the equivalence in the symplectic category that identifies twosymplectic manifolds M and N if M × Ck becomes equivalent to N × Ck for some k.

Having stabilized, we notice the appearance of structure relating to the categorical as-pects of geometric quantization. To begin with, we observe that the morphism spectrain our category are conjugate dual (see 4.3). In particular, there is an intersection pairingbetween stable lagrangians inside a symplectic manifold and those inside its conjugate.Endomorphisms of an object (M, ω) in our category can be seen as a homotopical notionof the “algebra of observables”. Our framework allows us to construct canonical stablerepresentations of this algebra (see 4.5, and 6.7). Extending coefficients on our category bya flat line bundle, we also recover a potential receptacle for the symbols of integral opera-tors in geometric quantization (see section 6). In addition, a derived version of geometricquantization can be realized as the 2-trace of this symbol (see 6.10).

Before we begin, let us say a word about our notation. In the following, given a pair ofsymplectic manifolds M, N , we shall construct two canonical bundles associated to thepair, which we shall denote by the letters τ and ζ . For the sake of simplicity, we willuse the same letters for different pairs of manifolds. The reader should interpret thosebundles in the context of the pair being considered.

We begin by thanking Gustavo Granja for his interest in this project and his hospitalityat the IST (Lisbon) where this project started. We thank Jack Morava for his continuedinterest and encouragement, and for sharing [9] with us. We would like to also thankDavid Ayala, John Klein and John Lind for helpful conversations related to various partsof this project. And finally, we would like to acknowledge our debt to Peter Landweberfor carefully reading an earlier version of this manuscript and providing several veryhelpful suggestions.

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2. STABLE LAGRANGIAN IMMERSIONS AND THEIR MODULI SPACE

In this section we will describe the stabilization procedure that we will apply to the sym-plectic category in later sections. To begin with consider a symplectic manifold (M2m, ω)endowed with a compatible almost complex structure. Let τ : M −→ BU(m) denotethe map that classifies its (complex) tangent bundle. We will assume that the cohomol-ogy class of ω is a multiple of the first Chern class of M . Now fix the model for BU(m)as the space of m-dimensional complex planes in C∞. In particular, the universal spaceEU(m) can be identified with the space of all orthonormal m-frames in C∞. We will chooseEU(m)/ O(m) as our model for BO(m). So we have a bundle BO(m) → BU(m) with fiberU(m)/ O(m). Consider the pullback diagram:

S(M)

ζ// BO(m)

Mτ

// BU(m)

Definition 2.1. Now let Xm be an arbitrary m-manifold, and let ζ be an m-dimensional realvector bundle over a space B. By a ζ-structure on X we shall mean a bundle map τ(X) −→ ζ ,where τ(X) is the tangent bundle of X .

Claim 2.2. The space of lagrangian immersions of X into M is equivalent to the space of ζ-structures on the tangent bundle of X , where ζ is the vector bundle over S(M) defined by theabove pullback.

Proof. A ζ-structure on τ(X) is the same thing as a map of X to M , along with an inclu-sion of τ(X) as an orthogonal (lagrangian) sub-bundle inside the pullback of the tangentbundle of M . Now the pullback of [ω] to H2(X, R) factors through H2(BO(m), R), by ourassumption that [ω] = c1 up to a scalar. Since H2(BO(m), R) = 0, the h-principle [6] maynow be invoked to show that the space of maps described above is equivalent to the spaceof lagrangian immersions of X in M .

Motivated by [7], consider the Thom spectrum S(M)−ζ . In a moment we will constructa family of such spectra and identify the stable spectrum given by their colimit. Geo-metrically, one may also describe a stabilization procedure on the moduli space of im-mersed lagrangians in M , and show how the infinite loop space for the former agreeswith the stabilized moduli space (see the discussion at the end of the section for a precisestatement). Let us provisionally observe that for a compact manifold X admitting a la-grangian immersion into M , an explicit stable map [X] : S0 −→ S(M)−ζ is constructed asthe composite:

[X] : S0 −→ X−τ(X) −→ S(M)−ζ

where the first map is the Pontrjagin–Thom construction on some embedding of X ⊂ R∞,and the next map is given by the (negative of the) ζ-structure on X .

Remark 2.3. Notice that the group of reparametrizations of the stable tangent bundle of X actson X−τ(X), and may potentially change the immersion class of X . We thank Thomas Kargh forpointing this out.

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Now if (M, ω) and (N, η) are two symplectic manifolds, the above construction yields acanonical map that represents the cartesian product of immersions:

µ : S(M)−ζM ∧ S(N)−ζN −→ S(M × N)−ζM×N

Taking (N, η) to be C with the standard symplectic structure, and including S−1 → S(C)−ζ

as the Thom spectrum over the point in S(C) represented by the line R ⊂ C we get:

Σ−1S(M)−ζM −→ S(M × C)−ζM×C

We may suspend this map to get a map:

S(M)−ζM −→ ΣS(M × C)−ζM×C

Definition 2.4. Define the Thom spectrum representing the space of stable lagrangian immersionsin M to be the colimit:

S(M)−ζ := hocolimk ΣkS(M × Ck)−ζk

where ζk denotes ζM×Ck . Alternatively, this spectrum can be seen as the Thom spectrum for thestable bundle ζ defined using the pullback:

S(M)

ζ// Z × BO

Mτ

// Z × BU

where τ is the stable tangent bundle of M of virtual (complex) dimension m.

Remark 2.5. Notice that by construction, we have a canonical equivalence:

S(M × C)−ζ = Σ−1 S(M)−ζ

where we have abused notation and used the same letter ζ to denote the stable bundle over S(M),and over S(M × C). We take this opportunity to introduce an abusive notation of using the sameletter ζ (regardless of the manifold) to mean the bundle defined by the above pullback. Hopefullythe bundle will be clear from the context of the manifold, and this notation will cause no confusion.

The product map stabilizes to a map:

µ : S(M)−ζ ∧ S(N)−ζ −→ S(M × N)−ζ

In the case of M = ∗, the spectrum S(∗)−ζ is the Thom spectrum of the (formal negative of the)stable vector bundle over (U/O) of virtual dimension zero given by the inclusion (U/O) → BO.In particular, S(∗)−ζ is a commutative ring spectrum, and the product map:

µ : S(∗)−ζ ∧ S(N)−ζ −→ S(N)−ζ

describes S(N)−ζ as a module over S(∗)−ζ .

Example 2.6. Consider the example of a cotangent bundle M = T ∗X , on a smooth m-dimensional manifold X and endowed with the canonical symplectic form. Now sincethe tangent bundle τ of M admits a lift to Z × BO, the space S(T ∗X) is homotopy equiv-alent to (U/O) × X . In particular, stably, a compact lagrangian immersion L → T ∗X isrepresented by a ζ × τ(X) structure on L, where ζ is the virtual zero dimensional bundleover (U/O) defined above. Hence the stable nature of caustics (defined as the critical setof the projection map π : L → X) is measured by a “universal Maslov structure” on L,

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which we define as a bundle map from the stable fiberwise tangent bundle of π, to thebundle ζ , that lifts the universal Maslov class L → U/O. Now it is easy to see that:

S(T ∗X)−ζ = S(∗)−ζ ∧ X−τ(X)

where X−τ(X) denotes the Thom spectrum of the formal negative of τ(X). If X were com-pact, then S(∗)−ζ ∧ X−τ(X) is equivalent to Map(X, S(∗)−ζ). This can be interpreted assaying that a stable lagrangian L in T ∗X can be interpreted as a family of virtual dimen-sion zero stable lagrangians parametrized as fibers of the map π : L → X .

The geometric meaning of the moduli space of stable lagrangians:

Let us briefly describe the geometric objects that our moduli space of stable immersionscaptures. Fix k > 0, and consider the Thom spectrum ΣkS(M×Ck)−ζk , where the notationis borrowed from the earlier part of this section. Using methods described in [3] (Sec.4.4), one can interpret the infinite loop space Ω∞−k(S(M × Ck)−ζk) as the moduli spaceof manifolds W m+k ⊂ R∞ × Rk, with a proper projection onto Rk, and endowed with alagrangian immersion W m+k −→ M × Ck. In this interpretation, the stable map:

ΣkS(M × Ck)−ζk −→ Σk+1S(M × Ck+1)−ζk+1

destabilizes to the map that simply sends W m+k to W m+k × R. Notice that the ”virtualdimension” m of the stable lagrangian is well defined. We thank David Ayala for patientlyhelping us understand this point of view.

Notice that for an (ordinary) lagrangian immersion L −→ M to represent a point in themoduli space Ω∞(S(M)−ζ), the manifold L must be compact. Assume now that L is a(possibly non-compact) manifold immersing into M . Then one may still construct a pointin the moduli space, provided one has a 1-form α on L×Rk for some k, with the propertythat α∗ιθ : L×Rk → Rk is proper. Here ιθ is the linear projection T ∗(L×Rk) → (Rk)∗ = Rk,which is being pulled back to L×Rk along α. In particular, any function φ on L×Rk so thatα = dφ is proper over Rk gives rise to a point in the stable moduli space. Such functionsare a natural analog of the theory of phase functions in our context [12].

3. THE STABLE SYMPLECTIC CATEGORY

Let us now describe the Stable Symplectic category S. By definition, the objects of this cat-egory S, will be symplectic manifolds (M, ω) (see remark 3.5). We will further assume thatwe have endowed each symplectic manifold with a compatible almost complex structure.In addition to this we assume for the sake of simplicity that (M, ω) is monotone, i.e. thatthe cohomology class of ω is c1(M) up to a scalar. 1

The morphisms in our category S will naturally have the structure of Thom spectra. Let(M, ω) and (N, η) be two objects. We define the conjugate of (M, ω) to be the symplecticmanifold M which has the same underlying manifold as M but with symplectic form −ω.

1It is possible to incorporate the case of arbitrary ω that admit integral representatives up to scalar.

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Definition 3.1. The “morphism spectrum” Ωζ(M, N) in S between M and N is defined as:Ωζ(M, N) := S(M × N)−ζ . We recall that S(M × N) is defined as the pullback:

S(M × N)

ζ// Z × BO

M × Nτ

// Z × BU

Assume now that M = N is a compact symplectic manifold. Then the diagonal lagrangian em-bedding ∆ : N → N × N induces a canonical isomorphism between TN ⊗ C and the restrictionof the tangent bundle of N × N along ∆. This defines a stable map [id] : S0 → Ωζ(N, N). Oncewe define composition, we shall see that [id] is indeed the identity in the category S.

The next step is to define composition. Consider k + 1 objects objects Mi with 0 ≤ i ≤ k,and let the space S(M0,...,k) be defined by the pullback:

S(M0,...,k)

ξi

// S(M 0 × M1) × · · · × S(Mk−1 × Mk)

M0 × ∆(Mi)k−1 × Mk

∆k−1

// M0 × (Mi × Mi)k−1 × Mk

Let ζi denote the individual structure maps S(M i−1 × Mi) −→ Z × BO, and let η(∆k−1)denote the normal bundle of ∆k−1. Performing the Pontrjagin–Thom construction alongthe top horizontal map in the above diagram yields a morphism of spectra:

Ωζ1(M0, M1) ∧ · · · ∧ Ωζk(Mk−1, Mk) −→ S(M0,...,k)

−λi

where λi : S(M0,...,k) −→ Z × BO is the formal difference of the bundle⊕

ζi and thepullback bundle ξ∗i η(∆k−1).

Now let π : S(M0,...,k) −→ M0×Mk denote the projection map onto the first and last factor.Then one checks that there is a canonical commutative diagram:

S(M0,...,k)

π

λi// Z × BO

M0 × Mk

τ// Z × BU

By the universal definition of S(M 0 ×Mk), this induces a canonical map from S(M0,...,k) toS(M 0 × Mk) that pulls ζ back to λi.

Definition 3.2. We define the composition map to be the induced composite:

Ωζ1(M0, M1) ∧ · · · ∧ Ωζk(Mk−1, Mk) −→ S(M0,...,k)

−λi −→ S(M0 × Mk)−ζ = Ωζ(M0, Mk).

We leave it to the reader to check that composition as defined above is homotopy associative. In fact,this composition has more structure. The question of how structured this associative compositionis will be addressed in a later document.

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Remark 3.3. We may interpret the above map geometrically as follows: Let Y = L1×L2×· · ·×Lk

be a product of stable lagrangians immersed in M0 × (Mi × Mi)k−1 × Mk. Assume that X ⊂ Y

is the transverse intersection of Y along M0 ×∆(Mi)k−1 ×Mk. Hence we get a factorization with

rows being given by the Pontrjagin–Thom construction:

Y −τ(Y ) //

X−τ(X)

Ωζ1(M0, M1) ∧ · · · ∧ Ωζk(Mk−1, Mk) // S(M0,...,k)

−λi

It is easy to see that X supports a lagrangian immersion into M0 × Mk, represented by the com-posite map:

[X] : S0 −→ X−τ(X) −→ S(M0,...,k)−λi −→ S(M 0 × Mk)

−ζ = Ωζ(M0, Mk).

Now let Ω∞ζ (M, N) denote the infinite loop space of the Thom spectrum. Then from the geometric

standpoint on the symplectic category, it is more natural to consider the unstable composition map:

Ω∞

ζ1(M0, M1) × · · · × Ω∞

ζk(Mk−1, Mk) −→ Ω∞

ζ (M0, Mk)

By applying π0 to this morphism, and invoking the above observation, we see that the definition ofcomposition is faithful to Weinstein’s definition of composition in the symplectic category.

Claim 3.4. Let M be a compact manifold, and let [id] denote the map [id] : S0 → Ωζ(M, M)corresponding to the diagonal embedding ∆ : M → M × M . Then [id] is indeed the identity forthe composition defined above.

Proof. Given two manifolds M, N , let ∆ ⊂ M ×M is the diagonal representative of [id] asabove. Observe that N ×∆(M)×M is transverse to N ×M ×∆ inside N ×M ×M ×M .They intersect along N ×∆3(M), where ∆3(M) ⊂ M ×M ×M is the triple (thin) diagonal.Hence we get a commutative diagram:

Ωζ(N, M) ∧ S0

))S

S

S

S

S

S

S

S

S

S

S

S

S

S

S

// Ωζ(N, M) ∧ ∆−τ(M) //

Ωζ(N, M) ∧Ω Ωζ(M, M)

Ωζ(N, M)=

// Ωζ(N, M)

where the right vertical map is composition, and the left vertical map is the Pontrjagin–Thom collapse over the inclusion map N × M = N × ∆3(M) −→ N × M × ∆. Nowconsider the following factorization of the identity map:

N × M = N × ∆3(M) −→ N × M × ∆ −→ N × M

where the last map is the projection onto the first two factors. Performing the Pontrjagin–Thom collapse map over this composite shows that the following composite is the identitymap:

Ωζ(N, M) ∧ S0 −→ Ωζ(N, M) ∧ ∆−τ(M) −→ Ωζ(N, M).

This shows that right multiplication by [id] : S0 → Ωζ(M, M) induces the identity map onΩζ(N, M). A similar argument works for left multiplication.

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Remark 3.5. The description of the identity morphism [id] : S0 → Ωζ(M, M) requires thatthe object M be compact. The price of extending S to include arbitrary symplectic manifolds asobjects, is that we simply lose the identity morphisms for non-compact objects. Notice that theconstruction of Ωζ(M, N) does not require M or N to be compact. Indeed, for arbitrary symplecticmanifolds M, N , the infinite loop space underlying Ωζ(M, N) is a valid moduli space of stableimmersed (compact) lagrangian submanifolds in M × N . Recall that we have a canonical equiva-lence: Ωζ(M × C, N) = Σ−1Ωζ(M, N). The same holds on replacing N by N × C. Therefore, upto natural equivalence, the morphism spectra factor through the equivalence on symplectic man-ifolds defined in the introduction. We will state explicitly when our manifolds are assumed to becompact.

4. INTERNAL STRUCTURE OF THE STABLE SYMPLECTIC CATEGORY

Let Ωζ(∗, ∗) be denoted by Ω, and we think of it as the “coefficient” spectrum. Fromthe previous section, we know that Ω = (U/O)−ζ is a commutative ring spectrum. Thisspectrum has been studied in [2] (Section 2). In particular,

π∗Ω = Z/2[x2i+1, i 6= 2k − 1].

In addition, the ring π∗Ω can be detected as a subring of π∗ MO. It follows that Ω is ageneralized Eilenberg–Mac Lane spectrum over H(Z/2). Since it acts on all morphismspectra Ωζ(M, N), we see that all the spectra Ωζ(M, N) are also generalized Eilenberg–Mac Lane spectra (see the Appendix). In addition, the composition map is defined in thecategory of Ω-module spectra. In the special case M = N , notice that Ωζ(M, M) is a ringspectrum 2. In general, this ring spectrum will not have a unit unless M is compact. Foran arbitrary pair M, N , the spectrum Ωζ(M, N) is a (left/right) module over the respectivering spectra. If M is a compact symplectic manifold, then we may smash with the identity[id] to obtain a ring map Ω → Ωζ(M, M). It can be shown that the action of Ω on Ωζ(M, N)factors through this map.

Remark 4.1. One has an oriented version of this category sS, obtained by replacing BO by BSO.All definitions go through in this setting verbatim. However, the ”coefficients” sΩ is no longera generalized Eilenberg–Mac Lane spectrum. Its rational homology (or stable homotopy) is easilyseen to be an exterior algebra:

π∗sΩ ⊗ Q = Λ(y4i+1)

The following results are easy consequences of the definitions:

Claim 4.2. Given arbitrary symplectic manifolds M and N , there is a natural decomposition ofΩζ(M, N) induced by the composition map:

Ωζ(M, ∗) ∧Ω Ωζ(∗, N) = Ωζ(M, N).

In particular, arbitrary compositions can be canonically factored, and computed by applying thefollowing composition map internally:

Ωζ(∗, N) ∧Ω Ωζ(N, ∗) −→ Ω.

Claim 4.3. Let M be a compact symplectic manifold. Then the spectrum Ωζ(M, ∗) is canonicallyequivalent to HomΩ(Ωζ(∗, M), Ω). In other words, Ωζ(∗, M) is conjugate dual.

2 See theorems 4.5 and 8.4 for an interpretation of Ωζ(M, M).

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Proof. Consider the composition map: Ωζ(∗, M) ∧ Ωζ(M, ∗) −→ Ω. Taking the adjoint ofthis map yields the map we seek to show is an equivalence:

Ωζ(M, ∗) −→ HomΩ(Ωζ(∗, M), Ω).

To construct a homotopy inverse to the above map, one uses the identity morphism:

HomΩ(Ωζ(∗, M), Ω) ∧ S0 −→ HomΩ(Ωζ(∗, M), Ω) ∧Ω Ωζ(M, M) −→ Ωζ(M, ∗),where the last map is evaluation, once we identify Ωζ(M, M) with Ωζ(M, ∗) ∧Ω Ωζ(∗, M)using the previous claim. Details are left to the reader.

Remark 4.4. This is really Poincare duality in disguise that says the the dual of the Thom spec-trum of a bundle ξ over M is the Thom spectrum of −ξ − τ , where τ is the tangent bundle of M .So the bundle that is “self dual” is the bundle ξ so that ξ = −ξ − τ , i.e. 2ξ = −τ . The bundle −ζis the universal bundle that satisfies this condition.

Theorem 4.5. For arbitrary symplectic manifolds M and N , there are canonical equivalences ofΩ-module spectra:

Ωζ(∗, M × N) = Ωζ(M × N, ∗) = Ωζ(N, M) = Ωζ(M, N).

Furthermore if M is compact, then using duality on M , we have an equivalence:

Ωζ(M, N) = HomΩ(Ωζ(∗, M), Ωζ(∗, N))

which is compatible with composition in S. In particular, for a compact manifold M , the ringspectrum Ωζ(M, M) has the structure of an endomorphism algebra 3:

Ωζ(M, M) = EndΩ (Ωζ(∗, M)) .

The above identification suggests that Ωζ(M, M) should be viewed as the homotopical “algebra ofobservables” of M . All this structure holds for sΩζ(M, M) as well. Furthermore, in the unorientedcase, the above theorem may be strengthened (see theorem 8.4).

5. THE METAPLECTIC CATEGORY

Kahler manifolds admit a holomorphic version of geometric quantization. This quanti-zation is the space of square integrable holomorphic sections of the line bundle given by

L ⊗√

det, where L is the prequantum line bundle, and√

det is a choice of square rootof the volume forms (called a metaplectic structure). A derived version of this quantiza-tion should in principle depend on all the cohomology groups of this line bundle. Nowthe Dolbeaut ∂-complex computing the cohomology of holomorphic bundles agrees withthe Spinc Dirac operator. On incorporating the metaplectic structure into the picture, thecomplex agrees with the Spin-Dirac operator. We therefore observe that almost complexSpin manifolds support a Dirac operator which generalizes the Dolbeaut complex twistedby the square root of the volume forms. This may suggest defining a derived version ofthe geometric quantization of a symplectic manifold with a Spin structure as the L2 in-dex of the Dirac operator with coefficients in the prequantum line bundle. For compact

manifolds this is simply the A genus with values in the prequantum line bundle. We takethis as good motivation to explore the nature of the stable symplectic category supportingsymplectic manifolds with this structure.

3In fact, we show in section 7 that the symplectomorphism group of M maps to the units in this algebra.

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In this section we will extend the constructions given in the previous sections to the meta-plectic category . The objects in this category will be symplectic manifolds endowed witha compatible metaplectic structure. Informally speaking, a compatible metaplectic struc-ture is a compatible complex structure endowed with a square root of the determinant linebundle. Let us now formalize the concept of a metaplectic structure. Let the classifying

space of the metaplectic group U, denoted by BU, be defined via the fibration:

BU −→ BU −→ K(Z/2, 2)

with the second map being the mod-2 reduction of the first Chern class. By definition, BU

supports the square-root of the determinant map√

det : BU −→ BS1. The complexifica-

tion map BSO −→ BU lifts to a unique map BSpin −→ BU. We may describe these lifts asa diagram of fibrations:

U/Spin

// U/SO

// K(Z/2, 1)×K(Z/2, 2)

BSpin //

BSO

w2// K(Z/2, 2)

0

BU // BUc1

// K(Z/2, 2)

An easy calculation shows that the mod-2 cohomology H∗(U/SO, Z/2) is an exterior alge-bra on generators σ, w2, w3, . . ., where σ is the class that transgresses to c1 in the mod-2Serre spectral sequence, and wi are the classes that are given by the corresponding re-strictions from H∗(BSO, Z/2). In addition, the class U/SO −→ K(Z/2, 1)×K(Z/2, 2) isrepresented by the product σ × w2. Hence U/Spin is the corresponding cover of U/SO.

Remark 5.1. The kernel of the map U → U/O is given by a group O called the Metalinear group

(see [12] Section 7.2). The classifying space BO is described by a fibration:

BO //

BO

w21

// K(Z/2, 2)

=

BU // BUc1

// K(Z/2, 2)

The group O can easily be seen as a split Z/4 extension of SO. The map BO → B(Z/4) is

sometimes called the Maslov line bundle. Notice also that BSpin can be seen as the cover of BOgiven by prescribing a trivialization of the Maslov line bundle, followed by a Spin structure.

We may now define the stable metaplectic category S along the same lines as before. The

objects of S are symplectic manifolds with a lift of their stable tangent bundle to BU. The

morphism spectra Ωζ(M, N) are defined as

Ωζ(M, N) := S(M × N)−ζ

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where ζ is the n + m dimensional bundle defined by the pullback:

S(M × N)

ξ

ζ// Z × BSpin

M × Nτ

// Z × BU

where τ denotes the 2(m + n)-dimensional tangent bundle of the product metaplecticmanifold M × N . Composition is defined analogously. As before, all structure maps in

the category are module maps over the coefficient spectra Ω = (U/Spin)−ζ .

6. SYMBOLS AND THOM CLASSES

Of particular interest in the theory of geometric quantization is the metaplectic category.Recall that one may motivate a derived version of quantization of a metaplectic mani-fold M as the L2-index of the Dirac operator with values in the prequantum line bundleL. This is motivated by the fact that for Kahler manifolds, this index is the holomorphicEuler characteristic of the prequantum line bundle twisted by the square root of the vol-ume forms. Under suitable vanishing conditions this Euler characteristic reduces to thespace of square integrable holomorphic sections, which is precisely the classical notionof the geometric quantization of a Kahler manifold. The propagators in geometric quan-tization are typically constructed as integral operators with kernels that are supportedon lagrangian submanifolds (see [12] Chapter 6). These kernels are flat sections of theprequantum line bundle restricted to the lagrangian submanifold. We shall explore suit-able scalar extensions of the metaplectic category along flat line bundles with the aim ofconstructing an algebraic receptacle for the propagators.

The category of symbols, and the symbol map:

In this section, we construct a family of categories S(B(~)), indexed by a nonzero parame-ter ~ ∈ R∗, which should be thought of the category of symbols for immersed lagrangiansin symplectic manifolds. This family admits a canonical projection to the trivial familyof symplectic categories S × R∗. We also construct a non-trivial section to this projection,which we call the symbol map. In what follows, we make our constructions with thesymplectic category S, with the understanding that all results hold for the categories sS

and S. Our constructions are arguably not as geometric as one would like; though theydo seem to suggest that there is perhaps an interesting underlying geometry.

Let B denote the classifying space of the group S1 considered as a discrete group. SoB = K(S1, 1). Notice that B classifies unitary line bundles with a flat connection, and onehas an extension of commutative topological groups:

K(R, 1) −→ B −→ BS1 .

This extension is classfied by a cocycle c1 ∈ H2(BS1, R) which is the universal first Chernclass with real coefficients. This is the restatement of the fact that a line bundle admits aflat connection if its curvature is an exact form. Any two choices of a flat connection ona line bundle over a base B differ by a closed form representing a class in H1(B, R). Wemay also define the extension B(~) to be the one classified by the cocycle 1

~c1.

11

Definition 6.1. For ~ ∈ R∗, define S(B(~)) to be the stable category with the same objects as S,and morphisms given by Ωζ(M, N) ∧ B(~)+. Since B(~) is a commutative topological group, itssuspension spectrum is a commutative ring spectrum. This allows us to define composition exactlyas before. All structure maps in the category S(B(~)) are defined in the category of Ω(B(~))-module spectra, where the coefficient spectrum Ω(B(~)) is the spectrum (U/O)−ζ ∧ B(~)+.

Let S×R∗ denote the trivial family of categories. Notice that there is a canonical projectionfunctor: ρ : S(B(~)) −→ S×R∗. The next step is to define a suitable section of ρ which onemay call the symbol σ for the symplectic category.

Fix a symplectic manifold M with a compatible complex structure. Assume ~ is a nonzeroconstant with the property that ~[ω] is an integral cohomology class. Fix a lift of ~[ω] toH2(M, Z), and let L(~) denote the “prequantum” line bundle classfied by that lift. Nowrecall our assumption that the cohomology class of the symplectic form ω is a scalar mul-tiple of the first Chern class of M . Hence the class ~[ω] ∈ H2(M, R) restricts trivially along

the map ξ (see below). Hence a lift L(~) exists:

S(M)

ξ

L(~)// B(~)

ML(~)

// BS1 .

Let us also make a choice of lift L(~) = L(~)(M) for the manifold M , and fix that choice

henceforth. Assume that the choice of the lift L(~)(M) for the conjugate manifold is given

by composing L(~)(M) with the formal negative map on B(~).

Consider the special case of the symplectic manifold C. Let us choose the trivial lift to

B(~) of the prequantum line bundle. One map extend it to the trivial lift L(~) of theprequantum line bundle Λk(Ck) over Ck. We denote by Lk(~) the prequantum line bundleL(~)⊗Λk(Ck) on M ×Ck. By the above constructions, we have a natural sequence of lifts

Lk(~) as shown below:

S(M × Ck)

ξ

Lk(~)// B(~)

M × CkLk(~)

// BS1 .

Composing Lk(~) with the diagonal map yields a sequence of natural maps of spectra:

σk(~)(M) : Σm+kS(M × Ck)−ζk −→ Σm+kS(M × Ck)−ζk ∧ B(~)+.

Define σ(~)(M) : S(M)−ζ −→ S(M)−ζ ∧ B(~)+ to be the colimit of this sequence.

Given two symplectic manifolds M and N with the property that ~[ω] is integral for both,the above construction gives us the symbol map on the spectra Ωζ(M, ∗) and Ωζ(∗, N).Now using the canonical factorization: Ωζ(M, ∗)∧ΩΩζ(∗, N) = Ωζ(M, N), we get a symbol:

σ(~)(M, N) : Ωζ(M, N) −→ Ωζ(M, N) ∧ B(~)+.

We leave it to the reader to check that this map respects composition.

12

Now define a category S(~−1Z) whose objects are pairs (M, ~), where M = (M, ω) is asymplectic manifold with the property that ~[ω] is endowed with an integral lift. Mor-phisms are defined as Ωζ(M, N) if (M, ~) and (N, ~) are objects with the same scalar ~.

Definition 6.2. Define the symbol functor σ : S(~−1Z) −→ S(B(~)) which sends each object toitself, and on morphisms, is given by:

σ(~)(M, N) : Ωζ(M, N) −→ Ωζ(M, N) ∧ B(~)+.

Remark 6.3. We point out that there were choices made in the construction of the symbol. Onemay parametrize the choices as a torsor over the 1-cocycles on S with values in the the cohomologygroups H1(M × N, R). Hence, one should interpret symbols as a notion of a flat connection onthe symplectic category.

Example 6.4. Returning to the example of M = T ∗X , for a compact manifold X . Anysymbol map is given by:

σ : Ω ∧ X−τ(X) −→ Ω(B) ∧ X−τ(X) = Map(X, Ω(B)).

This map destabilizes to the map that sends a stable lagrangian L to the parametrizedfamily of virtual dimension zero stable lagrangians along π : L → X , with values in aflat bundle. Of course, we are describing the push forward of the (flat) restriction of theprequantum line bundle on L, along the map π.

Thom classes:

The next item on the agenda is to construct Thom classes. Notice that the Thom spectraΩζ(M, N) admit canonical maps to Σ−(m+n) MO that classify the map induced by the vir-

tual bundle −ζ . Similarly, the oriented and metaplectic categories sS, S admit canonicalmaps to MSO and MSpin respectively. Continuing to work with S for simplicity, let E nowbe a spectrum with the structure of a commutative algebra over the spectrum MO. Thisallows us to obtain Ω-equivariant Thom classes:

E(M, N) : Ωζ(M, N) −→ Σ−(m+n) MO −→ Σ−(m+n) E .

Remark 6.5. Given a choice of Thom classes, notice that any object N in S has an induced Eorientation via the composite map N−τ → Ωζ(N, N) → Σ−2n E, induced by a lift of the diagonalinclusion N → N ×N . The multiplicative property of the Thom classes can be written as follows.By restricting −(ζ(M, N)⊕ ζ(N, P )) over M ×∆(N)×P , we have the equality of Thom classesgiven by: E(M, N) ∧ E(N, P ) = [N ] ∧ E(M, P ), where [N ] ∈ E−2n(N−τ ) is the N-orientation.

The Thom isomorphism theorem is now a formal consequence of the definitions. Con-sider the diagonal map: Ωζ(M, N) −→ Ωζ(M, N) ∧ (M × N)+, where (M × N)+ denotesthe space M × N with a disjoint base point. This allows us to construct Thom maps inE-homology and cohomology, given by capping with the Thom class, and cupping withit respectively. The Thom isomorphism now follows by an easy argument:

Claim 6.6. Let M and N be any two symplectic manifolds of dimension 2m and 2n resp. GivenΩ-equivariant Thom classes as above, there are canonical Thom isomorphisms:

π∗(Ωζ(M, N) ∧Ω E) = E∗+m+n(M × N), π∗ HomΩ(Ωζ(M, N), E) = E−∗+m+n(M × N).

The following is now an easy consequence of 4.5:

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Theorem 6.7. There exists a representation of the symplectic category in the category of π∗ E-modules:

q : E∗+m(M) ⊗ π∗Ωζ(M, N) −→ E∗+n(N).

In addition, one has an intersection pairing (which is non-degenerate for compact M):

E∗+m(M) ⊗ E∗+m(M) −→ E∗ .

This pairing, along with the above representation, induces a structure of a topological field theoryon the category S. All of this structure also holds for the oriented and metaplectic case.

Remark 6.8. Given an MO-algebra E as above, we may extend the category of symbols along

Ω(B(~)) = Ω ∧ B(~)+ −→ E∧B(~)+ := E(B(~)).

It follows from the Thom isomorphism that the symbol of a lagrangian L in M × N belongs toE(B(~))m+n(M×N). Furthermore, If M and N are assumed to be compact symplectic manifolds,then the symbol may be seen as an element in E(B(~))m+n(M × N) given by the composite:

M × N −→ Lτ(L) −→ Σm+n E(B(~)),

where the first map is the Pontrjagin–Thom collapse, and the second map is given by the Thomclass for τ(L) in E-cohomology, smashed with the prequantum line bundle on L.

Derived Geometric Quantization as a 2-trace of the symbol:

Motivated by the introductory discussion, let us now specialize to the case of the meta-

plectic category S. In this section, we shall extend coefficients along the MSpin moduleKU. Let us begin by recalling the symbol map from the previous section:

σ : S(~−1Z) −→ S(B(~)).

Notice that there is also the canonical “identity” functor that factors through the mapS0 −→ B(~)+:

Id : S(~−1Z) −→ S(B(~)).

Definition 6.9. A categorical 2-trace of σ is a natural transformation H : Id ⇒ σ. In other wordsH is given by a collection of maps H(M) : S0 −→ Ωζ(M, M) ∧ B(~)+, so that the followingdiagram commutes:

Ωζ(M, N)

H(M)∧σ

Id∧H(N)// Ωζ(M, N) ∧Ω Ωζ(N, N) ∧ B(~)+

∗

Ωζ(M, M) ∧Ω Ωζ(M, N) ∧ B(~)+∗

// Ωζ(M, N) ∧ B(~)+,

where the bottom horizontal, and right vertical maps are given by composition denoted by ∗.

It is easy to see from the definition that on the subcategory generated by compact objects,a canonical example of the 2-trace of the symbol is given by the map H(M) = [id]∩(1⊗L) :S0 −→ Ωζ(M, M) ∧ BS1

+. Notice that the line bundle 1 ⊗ L is not flat when restricted tothe diagonal lagrangian [id] : ∆ ⊂ M × M , and hence we are forced to take values in the(topological) line bundle BS1. We therefore have:

14

Theorem 6.10. Let S(BS1) denote the category S ∧ BS1+ supporting a canonical map from

S(B(~)). Then on extension to S(BS1), the class H(M) = [id] ∩ (1 ⊗ L) is a canonical 2-trace ofthe symbol on the subcategory generated by compact objects. In particular, working with the meta-

plectic category, and extending coefficients along the ring map Ω ∧ BS1+ −→ KU, the canonical

2-trace H(M) ∈ KU2m(M × M) may be identified with the derived quantization of M given bythe Dirac operator on M twisted by the prequantum line bundle L.

Proof. The only thing that requires proof is to show that on extending coefficients to KU,the 2-trace H(M) can be canonically identified with the derived quantization of M . Forthis, recall that the extension of H(M) is given by the composite:

S0 −→ M−τ(M) −→ Ωζ(M, M) ∧Ω KU = Σ−2m(M × M)+ ∧ KU,

where the first two maps represent M as the diagonal lagrangian in M×M twisted by theclass 1⊗L, and the last map is the Thom isomorphism. Projecting further to Σ−2m KU, we

get the A-genus of M twisted with L, which was our definition of derived quantization.

Remark 6.11. Consider the special case of M = N , and the map given by H(M) ∗ σ:

Ωζ(M, M) −→ Σ−2m KU∧(M × M)+.

One may restrict attention to the multiplicative monoid given by components of the moduli space:

Ω∞(Ωζ(M, M)) consisting of those stable lagrangian immersions L −→ M × M that map the(K-theory) fundamental class of L to that of the diagonal ∆. Then the image of the map H(M) ∗ σrestricted to this subspace is the class H(M) ∈ Ω∞+2m(KU∧(M × M)+). In particular, thelagrangians in these components “act” on the derived quantization H(M).

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7. A SECOND STABILIZATION

This section is very different in flavor from the previous sections. It is perhaps aimed at amore homotopical audience and the geometric reader may wish to skip it.

Working for simplicity with the category S, let (M, ω) be a compact object. Recall from 4.5that Ωζ(M, M) = EndΩ (Ωζ(∗, M)).

Now let GL Ωζ(M, M) = AutΩ (Ωζ(∗, M)) be defined as the components that induce in-vertible π0(Ω)-module maps in homotopy. Since Ωζ(∗, M) is a finite cellular Ω-module,we may define a ”second stabilization map”:

Q(M) : BGL Ωζ(M, M) −→ Ω∞K(Ω)

where K(Ω) denotes Waldhausen’s Algebraic K-theory spectrum of the commutative ringspectrum Ω. It is clear from the definition that the second stabilization can be defined for

the categories sS and S as well.

Remark 7.1. Notice also that the map Q(M) may be extended to THH(Ω):

Q(M) : BGL Ωζ(M, M) −→ Ω∞THH(Ω) = Ω∞ LB(U/O)−ζ = Ω∞((Sp/U)+ ∧ Ω),

where we have used [4] to identify THH(Ω). In the unoriented case this seems like a tractable mapto try and understand since we have a good handle on the spectra Ωζ(M, M). In addition, we alsoknow the homotopy of THH(Ω):

π∗THH(Ω) = Z/2[x2i+1, y4j+2, i 6= 2k − 1].

Let us now concentrate on the oriented category sS. The spectrum sΩ is a connectivespectrum with π0sΩ = Z. Let us consider the fibration:

K(π) −→ K(sΩ) −→ K(Z)

where π is the fiber of the zero-th Postnikov section: sΩ → HZ.

Claim 7.2. Let K(sΩ) denote the cofiber of the canonical map K(S0) → K(sΩ). Then rationally,the spectrum K(π) is equivalent to K(sΩ). In particular the above fibration admits a canonicalrational splitting, and there exist polynomial classes y4i+2 in degree 4i + 2 such that π∗K(π) isisomorphic to the augmentation ideal:

π∗K(π) ⊗ Q = Q[y4i+2]>0.

Furthermore, rationally π∗K(π) can be identified with the injective image of the canonical map:

Ω∞Σ∞(BGL1(sΩ)) −→ BGL∞(sΩ)+ = Ω∞K(sΩ)

Proof. Since K(S0) is rationally equivalent to K(Z), the first part of the claim in clear.Now, via the Thom isomorphism, we may identify sΩ rationally with the ring spectrumΣ∞(U/SO)+. In particular π∗sΩ ⊗ Q = Λ(y4i+1). Now invoking results from [1], we seethat π∗K(π) can be identified with positive degree elements in THH∗(U/SO+) that are inthe kernel of the Connes boundary operator. These elements are given by the augmenta-tion ideal in H∗(B(U/SO)) = Q[y4i+2]. The proof of the claim is complete once we observethat the rational equivalence between sΩ and Σ∞(U/SO)+ induces a rational equivalencebetween BGL1(sΩ)) and B(U/SO).

16

Now given a compact symplectic manifold (M, ω), there exists a homomorphism fromthe symplectomorphism group to the units GL sΩζ(M, M) given by taking a symplecto-morphism to its graph:

BGr : BSymp(M, ω) −→ BGL sΩζ(M, M)

To explicitly construct this map observe that we may construct the spectrum sΩζ(∗, M)fiberwise over BSymp(M, ω). More precisely, consider the space J(M) consisting of pairs(J, m) with m ∈ M and J a compatible complex structure on (M, ω). This space fibersover the space of compatible complex structures on (M, ω), with fiber M . In particular,it is homotopy equivalent to M . There is a natural action of the symplectomorphismgroup on J(M), and the total space of the associated bundle ESymp(M, ω) ×Symp J(M)supports a canonical complex vector bundle extending the fiberwise bundle TM . Sincewe may replace J(M) by M without changing the homotopy type, the space given by

M = ESymp(M, ω) ×Symp M supports a complex vector bundle extending the tangentbundle of M .

Consider the pullback sS(M) fibering over BSymp(M, ω):

BSymp(M, ω)

=

sS(M)oo

ξ

ζ// Z × BO

BSymp(M, ω) Mooτ

// Z × BU

In particular, the fiberwise Thom spectrum sS(M)−ζ is a bundle of sΩ-module spectraover BSymp(M, ω), with fiber being sΩζ(∗, M). We may classify the bundle by:

BGr : BSymp(M, ω) −→ BAutsΩ sΩζ(∗, M) := BGL sΩζ(M, M)

We may now consider the K(sΩ)-valued parametrized index defined as the composite:

I(M) = Q(M) BGr : BSymp(M, ω) −→ Ω∞K(sΩ)

By [5], the map I(M) has a factorization:

BSymp(M, ω) −→ Ω∞Σ∞(M+) −→ Ω∞A(M) −→ Ω∞K(sΩ)

where A(M) is the Waldhausen K-theory of M , the first map is the Becker–Gottlieb trans-fer, the second is the natural transformation between Σ∞(X+) and A(X), and the final

map is the one that classfies the stable bundle of sΩ-spectra over M .

Using naturality, we have the following description of I(M) given by the commutativediagram with the vertical maps begin induced by the map that classifies the (rank one)

17

bundle of sΩ-spectra over M :

Σ∞(BSymp(M, ω)+)tr

//

=

Σ∞(M+)

// A(M) //

K(sΩ)

=

Σ∞(BSymp(M, ω)+) //

=

Σ∞(BU+) //

A(BU) //

K(sΩ)

=

Σ∞(BSymp(M, ω)+) // Σ∞(BGL1(sΩ)+) // A(BGL1(sΩ)) // K(sΩ)

Let us now attempt to understand the map BU −→ Ω∞Σ∞(BGL1(sΩ)+), given by ap-plying Ω∞Σ∞ to the map BU −→ BGL1(sΩ) that classifies the bundle of sΩ-spectra overBU.

Theorem 7.3. Let λ : BU −→ BGL1(sΩ) be the map above. Then its image in cohomology is thepolynomial algebra generated by all the odd Newton polynomials N2k+1(c1, c2, . . . , c2k+1) in theuniversal Chern classes.

Remark 7.4. We remind the reader that the Newton polynomials Ni(σ1, σ2, · · · , σi) are definedby (uniquely) writing the power symmetric functions xi

1 +xi2 + · · ·+xi

i in terms of the elementarysymmetric functions σ1, σ2, · · · , σi.

Proof. Recall that rationally H∗(GL1(sΩ)) is identified with H∗(U/SO+). Furthermore, itis an exterior algebra on generator in degree 4k + 1, and the map induced on the levelof groups by Ω(λ) : U → GL1(sΩ) is rationally surjective. It follows that the mapλ : BU −→ BGL1(sΩ) picks out the collection of polynomial cohomology generators indegree 4k + 2. Now since λ can be delooped, the cohomological generators can be chosento be primitive. It is a standard fact that the only primitive in degree 2i is the Newtonpolynomial Ni(c1, c2, · · · , ci).

The following result follows from the above theorem and claim 7.2.

Theorem 7.5. The parametrized index map I(M) can be identified with the map given by apply-

ing the Becker–Gottlieb transfer: [tr] : H∗(M) −→ H∗(BSymp(M, ω)) to a sub-algebra inside

H∗(M). This sub algebra is generated by the odd Newton polynomials in the Chern classes of the

fiberwise tangent bundle of M −→ BSymp(M, ω).

Remark 7.6. Notice that the claim 7.2 shows that we have a rational pullback diagram:

Σ∞(BGL1(sΩ)+)

// S0

K(sΩ) // K(Z)

Hence we may define secondary rational (Reidemeister) invariants in dimensions 4k, for familiesof symplectic manifolds (M, ω) that admit a prescribed null homotopy for the parametrized index.

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8. APPENDIX: SOME COMPUTATIONS AND QUESTIONS

The Unoriented Case:

Let us make some explicit computations in the case of the unoriented symplectic cate-gory S. We invoke the Adams spectral sequence to compute π∗Ωζ(M, N). Since Ωζ(M, N)is a generalized Eilenberg–Mac Lane spectrum, the spectral sequence will collapse andwe simply need to compute the primitives under the action of the dual mod-2 Steenrodalgebra on H∗Ωζ(M, N) := H∗(S(M × N)−ζ).

Remark 8.1. Let us set some notation. All homology groups will be understood to be over Z/2.In addition, let us use the suggestive notation Σ−ζS∗ to denote the shift Σ−(m+n)S∗ for a gradedmodule S∗.

Theorem 8.2. π∗Ωζ(M, N) is a free π∗Ω-module on a (non-canonical) generating vector spacegiven by Σ−ζH∗(M × N).

Proof. The Thom isomorphism theorem implies that the ring H∗Ω is isomorphic to H∗(U/O).Under this isomorphism we also see that H∗Ωζ(M, N) is isomorphic to Σ−ζH∗(S(M ×N)).Now consider the universal fibration U/O → BO −→ BU. It is easy to see that theSerre spectral sequence in homology for this fibration collapses leading to the fact thatH∗(S(M × N)) is non-canonically a free H∗(U/O)-module on H∗(M × N). From this wededuce that H∗Ωζ(M, N) is free H∗Ω-module on the (non-canonical) vector space givenby Σ−ζH∗(M × N). An easy argument using the degree filtration shows that the gener-ating vector space can be chosen to have trivial action of the dual Steenrod algebra. Thestatement of the theorem is now complete on taking primitives under this action.

The following consequences of the above theorem are easy (compare with 4.5):

Theorem 8.3. There is a natural decomposition of π∗Ωζ(M, N) induced by the composition map:

π∗Ωζ(M, ∗) ⊗π∗Ω π∗Ωζ(∗, N) = π∗Ωζ(M, N).

In particular, arbitrary compositions can be canonically factored in homotopy, and computed byapplying the composition map internally:

π∗Ωζ(∗, N) ⊗π∗Ω π∗Ωζ(N, ∗) −→ π∗Ω.

Theorem 8.4. Given a compact manifold M , the π∗Ω-algebra π∗Ωζ(M, M) has the structure ofan endomorphism algebra:

π∗Ωζ(M, M) = Endπ∗Ω (π∗Ωζ(∗, M)) .

The Oriented Case:

Next, let us very briefly explore the structure of sΩζ(∗, M) rationally. Firstly recall thatH∗(U/SO, Q) is an exterior algebra Λ(y4i+1). Now by Thom isomorphism, we have anequality H∗+m(sS(M), Q) = H∗(sΩζ(∗, M), Q), where (M, ω) is a 2m-dimensional mani-fold. Now consider the cohomology Serre spectral sequence for the fibration

U/SO −→ sS(M) −→ M.

19

It is easy to see that the class y1 ∈ H1(U/SO, Q) transgresses to a non-trivial multipleof the symplectic class ω. Hence the class y1 ∪ ωm represents the only meaningful pri-mary characteristic class in H2m+1(sS(M), Q). Let θ(M) be the corresponding class inHm+1(sΩζ(∗, M), Q) under the above Thom isomorphism.

Now let π : E → B be a fibrating family of oriented stable lagrangians in M , endowedwith a classifying map f(π) : B → Ω∞(sΩζ(∗, M)). Then the map f(π) factors throughthe Umkehr map B+ → Ω∞(E−τ(π)) followed by the map induced by E−τ(π) → sS(M)−ζ .It follows that

f(π)∗θ(M) = π∗(y1 ∪ ωm),

where y1 ∪ωm denotes the pullback of the class having the same name along E → sS(M).

Some Interesting questions:

Here is a list of natural questions, some of which are quite vague or ambitious.

Question 8.5. Is there a universal description of the stable symplectic or metaplectic cat-egory that allows us to check if a functor defined on symplectic manifolds extends tothe stable category? Notice that if F(M) denotes any (stable) representation of S, withF(∗) := F being an Ω-module, then we have the action map for F on the level of spectra:

q : Ωζ(M, ∗) ∧Ω F(M) −→ F .

For compact manifolds M , we may dualize this map to get a “symbol”:

F(M) −→ Ωζ(∗, M) ∧Ω F .

Question 8.6. Describe the “Motivic Galois group”, by which we mean the rule that as-signs to a commutative Ω-algebra F, its F-points given by the group of multiplicative au-tomorphisms of the monoidal functor on S (with values in the category of π∗ F-modules),and which takes a symplectic manifold N to π∗(Ω(∗, N) ∧Ω F). In the oriented case ofsΩ, one may wish to ask the question about multiplicative automorphisms of the rationalfunctor with F-points given by π∗(Ω(∗, N) ∧Ω F) ⊗ Q.

Question 8.7. Recall the homomorphism from the symplectomorphism group to the unitsGL Ωζ(M, M) given by taking a symplectomorphism to its graph:

BGr : BSymp(M, ω) −→ BGL Ωζ(M, M)

We showed that the above map BGr classifies the (twisted canonical) fibration:

S(M)−ζ := ESymp(M, ω)+ ∧Symp Ωζ(∗, M) −→ BSymp(M, ω)

It can be shown that BGr is null if and only if the Ω-based (Atiyah–Hirzebruch) Serrespectral sequence for the above fibration collapses. It would be very interesting to knowthe kind of invariants the map BGr detects.

Question 8.8. Are the degree 4n + 1 classes in π∗K(sΩ) (detected in π∗K(Z)) related toπ4n BSympc(C

k), for large values of k and where Sympc denotes compactly supportedsymplectomorphisms?

Question 8.9. Is there a natural geometric structure that makes the symplectic categorywith coefficients in KU(B(~)), into a “KU-linear” stable category? In particular, is there ageometric interpretation of the results in section 6?

20

REFERENCES

1. C. Ausoni and J. Rognes, Rational Algebraic K-theory of topological K-theory, available atarXiv:math.AT/0708.2160v1, 2007.

2. V. I. Arnold and A. B. Givental, Symplectic geometry, in Dynamical systems, IV, Encyclopaedia Math.Sci., 4, Springer, Berlin, 1–138, 2001.

3. D. Ayala, Geometric cobordism categories, available at arXiv:0811.2208.4. A. Blumberg, R. Cohen and C. Schlichtkrull, Topological Hochschild homology of Thom spectra and the free

loop space, Geom. Topol. 14, No. 2, 1165–1242, 2010.5. W. Dwyer, M. Weiss and B. Williams, A prarmetrized Index theorem for the algebraic K-theory Euler class,

Acta Math. 190, 1–104, 2003.6. Y. Eliashberg and N. Mishachev, Introduction to the h-principle, AMS, Graduate Studies in Mathematics,

Vol. 48, 2002.7. S. Galatius, I. Madsen, U. Tillman and M. Weiss, The Homotopy Type of the Cobordism Category. Acta

Math. 202, no. 2, 195–239, 2009.8. V. Guillemin and S. Sternberg, Some problems in integral geometry and some related problems in microlocal

analysis, Amer. J. Math. 101, 915–955, 1979.9. J. Morava, Smooth correspondences, Prospects in topology, Ann. of Math. Stud., 138, Princeton Univ.

Press, Princeton, NJ, 270–282, 199510. A. Weinstein, Symplectic Categories, available at arXiv:0911.4133.11. S. Bates and A. Weinstein, Lectures on the Geometry of Quantization, Berkeley Mathematical Lecture Notes,

1997. Available at http://math.berkeley.edu/ alanw/.12. A. Weinstein, Symplectic Geometry, Bulletin of the A.M.S., Vol.5, No.1, 1–13, 1981.13. K. Wehrheim and C. Woodward, Functoriality for Lagrangian correspondances in Floer theory, available at

arXiv:0708.2851.

DEPARTMENT OF MATHEMATICS, JOHNS HOPKINS UNIVERSITY, BALTIMORE, USA

E-mail address: nitu@math.jhu.edu